I have a somewhat unrelated question: is there any evidence that Neanderthals couldn't articulate certain consonants or vowels that humans are capable of, or any consonants/vowels that humans cannot articulate? Is there even any evidence that could guide a sort of shot-in-the-dark "realistic" portrayal of Neanderthal language?
That's tricky to know, since we don't know what Neanderthal tongues or lips looked like.
We do have a couple bits of potentially useful information, though. For example, Neanderthals lacked a chin protuberance, which serves as a muscle attachment point for the lower lip. The lack of this protuberance may have restricted Neanderthals' ability to use bilabial (b, p) and labiodental (f) consonant sounds. Maybe.
Also,
this article has a link to a simulation of a Neanderthal voice pronouncing the letter "e". I have no idea how they came up with this simulation, but, since they were expert linguists and anthropologists, I think it's safe to assume that they had some kind of system worked out. The article also has some other tidbits on Neanderthal language possibilities, but you're going to have to dig further than this if you want to realistically reconstruct Neanderthal language.
I wouldn't blame you if you wanted to just make some stuff up.